Suppose that either of two instruments might be used for making a certain measurement. Instrument 1 yields a measurement whose p.d.f. is
f1(x)=2x, 0 <x<1

Instrument 2 yields a measurement whose p.d.f. is
f2(x)=3x^2, 0 <x<1

Suppose that one of the two instruments is chosen at random and a measurement X is made with it.
(a)
Determine the marginal p.d.f. of X.
(b)
If X = 1/4 what is the probability that instrument 1 was used?
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Now, part (a) asks f1(x) [function associated with probability that X=x] which we can compute with g1 and f2; part (b) asks g2(y|x) [function associated with probability that Y=y, given that X=x] which we can compute with g1, f2, and the in part (a) found f1.